A Simple Analytical Model for Rocky Planet Interiors

Li Zeng1,a and Stein B. Jacobsen1,b

1Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138

ABSTRACT

This work aims at exploring the scaling relations among rocky exoplanets. With the assumption of internal gravity increasing linearly in the core, and stay- ing constant in the mantle, and tested against numerical simulations, a simple model is constructed, applicable to rocky exoplanets of core mass fraction (CMF) ∈ 0.1 ∼ 0.4 and mass ∈ 0.1 ∼√10M⊕. Various scaling relations are derived: (1) 2 core radius fraction CRF ≈ CMF, (2) Typical interior pressure Ptypical ∼ gs 1 (surface gravity squared), (3) core formation energy Ediﬀ ∼ 10 Egrav (the total Mp 27 gravitational energy), (4) eﬀective heat capacity of the mantle Cp ≈ ·7·10 M⊕ −1 1 2 JK , and (5) the moment of inertia I ≈ 3 · Mp · Rp. These scaling relations, though approximate, are handy for quick use owing to their simplicity and lu- cidity, and provide insights into the interior structures of rocky exoplanets. As examples, this model is applied to several planets including Earth, GJ 1132b, Kepler-93b, and Kepler-20b, and made comparison with the numerical method.

Subject headings: Earth - planets and satellites: composition - planets and satellites: fundamental parameters - planets and satellites: interiors - planets and satellites: terrestrial planets

1. Introduction

Masses and radii of rocky exoplanets have been found for about a dozen cases (Figure 1), including Kepler-21b (Howell et al. 2012; Lopez-Morales et al. 2016), Kepler-20b (Gautier et al. 2012; Buchhave et al. 2016), COROT-7b (Leger et al. 2009; Queloz et al. 2009; Hatzes et al. 2011; Wagner et al. 2012; Haywood et al. 2014; Barros et al. 2014), HD219134b (Vogt et al. 2015; Motalebi et al. 2015), Kepler-10b (Batalha et al. 2011; Wagner et al. 2012; Dumusque et al. 2014; Esteves et al. 2015), Kepler-93b (Ballard et al. 2014; Dressing et al. – 2 –

2015), Kepler-36b (Carter et al. 2012; Morton et al. 2016), Kepler-78b (Pepe et al. 2013; Grunblatt et al. 2015), K-105c (Rowe et al. 2014; Holczer et al. 2016; Jontof-Hutter et al. 2016), GJ 1132b (Berta-Thompson et al. 2015; Schaefer et al. 2016), and more are likely. Here we explore what other parameters can be further gleaned from this information. Our earlier work (Zeng et al. 2016) shows that by using an equation of state (EOS) for Earth for diﬀerent core mass fractions (CMFs), a simple relationship between CMF, planetary radius, and mass can be found as

" # 1 R M 0.27 CMF = · 1.07 − p / p (1) 0.21 R⊕ M⊕

This work shows that the CMF can be related to the core radius fraction (CRF) of a rocky planet. A simple structural model can be calculated, which depends only on three parameters, (1) surface gravity gs, (2) planet radius Rp, and (3) core radius fraction CRF. The procedure is as follows:

G·Mp 1. Surface gravity gs = 2 can be calculated from mass Mp and radius Rp of a rocky Rp planet, or directly from combining transiting depth with radial-velocity amplitude (Equation (16)).

2. CMF can be determined from Equation (1). √ 3. CRF can be estimated as CMF.

The only assumption of this model is that the internal gravity proﬁle can be approximated as a piecewise function (see Figure 2 Panel (1)a-d):

1. In the core, the gravity g increases linearly with radius from 0 at the center to g s (surface value) at the core-mantle boundary (CMB): g (r) = g · r ∝ r core s Rcore

2. In the mantle, g stays constant: gmantle(r) = gs = const

This assumption is equivalent to assuming constant core density, followed by the mantle 2 1 density decreasing to 3 of the core density at CMB, and density deceasing as r in the mantle. The validity of this assumption is tested against the numerical results from solving the planetary structures with realistic EOS derived from PREM (Dziewonski & Anderson 1981), across the mass-radius range of 0.1∼10 M⊕ and 0.1.CMF.0.4 for two-layer (core+mantle) – 3 –

2.0 2.0 ) O 2 O O H 2 2 (rock % % % 25 1.8 100 1.8 MgSiO

Fe % 25 Fe % 1.6 50 1.6 ) ⊕ R (

r 1.4 1.4 Fe % 100

1.2 1.2

1.0 1.0

0.8 1 1.5 2 3 4 5 6 7 8 9 10

m (M⊕)

Fig. 1.— Mass-radius plot showing selected rocky planets. Curves show models of different compositions, with solid indicating single composition (Fe, MgSiO3, i.e. rock, H2O) and dashed indicating Mg-silicate planets with different amounts of H2O or Fe added. Rocky planets without volatile envelope likely lie in the shaded region within uncertainty, and those ones with volatile envelope may lie above. Planets are color-coded by their incident bolometric stellar flux (compared to the Earth) and equilibrium temperatures assuming (1) circular orbit (2) uniform surface temperature (3) bond albedo A=0. Earth and Venus are shown for reference. – 4 –

Fig. 2.— Numerical calculations based on PREM-extrapolated EOS (black) versus simple analytical models: Core (red, purple, and pink-area in between) and Mantle (green). Panel (1-4)a: Earth, Panel (1-4)b: GJ 1132b, Panel (1-4)c: Kepler-93b, Panel (1-4)d: Kepler-20b. Panel (1)a-d: Gravity Profiles (core is proportional to r and mantle is constant). Panel (2)a-d: Density Profiles (core is constant and mantle is inversely proportional to r). Panel (3)a-d: Pressure Profiles (core is parabolic in r and mantle is logarithmic in r). Panel (4)a- d: Temperature Profiles (best estimates shall lie in the green area (mantle) and pink area (core)). The solidus (where mixture starts to melt) and liquidus (where mixture completely melts) are plotted for comparison. – 5 – rocky planets. Note that Mercury lies outside this range of the CMF as it has a big core, owing to its likely giant impact origin (Asphaug & Reufer 2014). Various scaling relations are derived from this model.

2. Scaling Relation between Pressure and Gravity

The two ﬁrst-order diﬀerential equations (Seager et al. 2007; Zeng & Seager 2008; Zeng & Sasselov 2013) governing rocky planet interiors are 1. a hydrostatic equilibrium (force balance) equation: dP Gmρ = − = −g · ρ (2) dr r2

2. a mass conservation equation: dm = 4πr2ρ (3) dr

Equations (2) and (3) can be combined to give a relation between the internal pressure P and the mass m (mass contained within radius r, now used as the independent variable instead):

dP Gm 1 g2 1 d ln(m) = − = − · = − · g2 · (4) dm 4πr4 4πG m 4πG dm Integrating Equation (4), we get (ln(m) stands for the natural logarithmic of m)

Z interior 1 Z mass enclosed inside dP = − g2 · d ln(m) (5) surface 4πG Mp

This integration is from the surface inward, as the pressure at the surface is zero. Therefore, the typical internal pressure is on the order of

g2 P ∼ (6) 4πG

2 2 where g is some average of g . Deﬁning planet mass as Mp, planet radius as Rp, and planet mean density as ρp, then surface gravity gs and characteristic interior pressure Ptypical are – 6 –

GMp gs ≡ 2 (7) Rp

2 2 gs GMp Ptypical ≡ = 4 (8) 4πG 4πRp

Ptypical will be shown to approximate PCMB (pressure at CMB) later on. Given gs in S.I. units (m s−2) and pressures in GPa:

2 PCMB ∼ Ptypical ∼ gs (9)

−2 2 For example, g⊕ (Earth’s gravity)≈ 10m s , and g⊕ ≈ 100 is near P⊕,CMB = 136 GPa. The values of other planets are listed in Table 1.

3. Density Proﬁle

Based on the assumption of the gravity proﬁle, the density proﬁle is

3gs ρp 3 ρcore(r) = = = constant, thus, mcore(r) ∝ r (10a) 4πGRcore CRF

g 1 ρ (r) = s ∝ , thus, m (r) ∝ r2 (10b) mantle 2πGr r mantle

1 Figure 2 Panel (2)a-d compares this to the PREM-derived density proﬁles. The r dependence approximates the compression of mantle material toward depth, and the smaller core (CMF .0.4) allows the core density to be approximated as constant. Generally, Equa- tion (10a) approximates the density of the core near the CMB. Anywhere in the mantle,

m r 2 = (11) Mp Rp

In particular, at the CMB,

M R 2 CMF = core = core = CRF2 (12) Mp Rp – 7 –

In reality, this exact relation (Equation (12)) becomes approximate: √ CRF ≈ CMF (13)

The error of Equation (13) is generally within ∼10% (see Table (1)). It is a quick way to estimate the CRF from the CMF and vice versa. It can even be applied to a rocky planet with a volatile envelope if it is only applied to the solid portion of that planet.

Table 1: Calculated parameters for four planets from this simple analytical model Earth GJ 1132b Kepler-93b Kepler-20b

M(M⊕) 1 1.62 4.02 9.70 R(R⊕) 1 1.16 1.478 1.868 CMF (Equation (1)) 0.33 0.25 0.26 0.28 1 CMFN 0.32 0.27 0.28 0.22 CRF (Equation (13)) 0.58 0.50 0.51 0.53

CRFN 0.55 0.49 0.49 0.44 −3 ρp(g cm ) 5.5 5.7 6.9 8.2 −2 gs(m s ) (Equation (7)) 9.8 12 18 27 2 Ptypical (TPa ) (Equation (8)) 0.12 0.17 0.4 0.9 PCMB (TPa) (Equation (15)) 0.13 0.23 0.5 1.1 P0 (TPa) (Equation (22) 0.3 0.5 1.1 2.5

Pf0 (TPa) (Equation (23)) 0.40 0.56 1.4 3.6 32 Egrav(10 J) (Equation (27)) 2.4 5.6 27 120 32 Ediﬀ (10 J) (Equation (28)(29)) 0.2 0.5 2.4 11 thermal 32 Emantle (10 J) (Equation (33)(34)) 0.1 0.2 0.5 1.1 3 Tmp(10 K) 1.6 1.6 1.6 1.6 CMB 3 Tmantle(10 K) 2.5 3.1 3.6 4.1 CMB 3 Tcore (10 K) 4 5 8 11 3 Tcenter(10 K) 6 7 12 20 I(1038kg m2) (Equation (39)(40)) 0.8 1.8 7.2 28

4. Pressure Proﬁle

4.1. Pressure in the Mantle

Integrating Equation (5) with constant mantle gravity, we obtain

1N stands for numerical simulation using ManipulatePlanet at astrozeng.com 2TPa = 1000 GPa = 1012 GPa. – 8 –

g2 M M R P (m) = s · ln p = P · ln p = 2P · ln p (14) mantle 4πG m typical m typical r

Evaluating Equation (14) at the CMB gives PCMB (pressure at the CMB):

1 g2 1 P = P · ln = s · ln (15) CMB typical CMF 4πG CMF

For CMF ∈ 0.1 ∼ 0.4, PCMB ∈ (0.9 ∼ 2.3)Ptypical.

PCMB is an important physical parameter, as it determines the state of core and mantle materials in contact. Equation (15) only depends on gs and CMF. And gs can be determined independent of the stellar parameters (Southworth et al. 2007) as

2 1/2 2π (1 − e ) ARV gs = 2 (16) Porb (R/a) Sin[i] where semi-amplitude ARV and orbital eccentricity e can be constrained from the radial- velocity curve, and R/a is the radius over the semi-major axis ratio, which could be constrained directly from the transit light curve. The orbital period Porb can be constrained from both. Thus, it is possible to estimate PCMB even without knowing the accurate mass and radius in some cases for rocky planets.

4.2. Range of Applicability of This Model

From a theoretical point, we explore the range of applicability of this model.

∂P Bulk modulus K ≡ ∂ ln(ρ) . Therefore, in the mantle,

2 ∂Pmantle gs d ln(m) Kmantle = = · = 2 · Ptypical (17) ∂ ln(ρmantle) 4πG d ln(r)

Thus, in this model, the bulk modulus is constant everywhere in the mantle, equal to twice the typical internal pressure Ptypical. Realistically, K shall increase with pressure, so how good is this approximation?

2 g⊕ For Earth, P⊕,typical = 4πG = 115 GPa, so K⊕,mantle = 230 GPa. Comparing it to the – 9 –

 K⊕,LID = 130 GPa  isentropic bulk modulus Ks of Earth’s mantle according to PREM: K⊕,670km = 255.6 ∼ 300 GPa  K⊕,D00 = 640 GPa

So K⊕,mantle represents the midrange of the realistic bulk modulus in the mantle. For higher masses, let us invoke the BM2 (Birch-Murnaghan second-order) EOS (Birch 1947, 1952), which when ﬁtted to PREM gives K0 ≈ 200GPa for both core and mantle (Zeng et al. 2016):

" 7 5 # 3 ρ 3 ρ 3 P = · K0 − (18) 2 ρ0 ρ0

Again, K is obtained by diﬀerentiating Equation (18):

" 7 5 # ∂P 3 7 ρ 3 5 ρ 3 K ≡ = · K0 · − · (19) ∂ln(ρ/ρ0) 2 3 ρ0 3 ρ0

( when P K ,K ≈ K Equation (19) suggests: . 0 0 . Since P is the typical pres- 7 typical when P K0,K → 3 P ≈ 2P sure in the mantle, K ≈ 2P ≈ 2Ptypical. It is the same as Equation (17). Therefore, this approximation shall exist for higher masses as long as BM2 EOS holds. BM2 EOS’s validity

range extends above 10 M⊕, and here we set the upper limit to be 10 M⊕.

4.3. Pressure in the Core

Since ρcore = constant in this approximation, from Equation (2) we have

dPcore(r) G gs 3 3gs 3r = − 2 · r · = − 2 · Ptypical (20) dr r GRcore 4πGRcore Rcore Integrating it gives the pressure dependence on the radius as a parabolic function:

3 r 2 Pcore(r) = P0 − · Ptypical · (21) 2 Rcore

P0 (central pressure) can be determined by connecting Equation (21) at CMB to PCMB from Equation (15) as – 10 –

3 1 3 P = P + P = P · ln + ⇒ P ∈ (2.4 ∼ 3.8) · P (22) 0 CMB 2 typical typical CMF 2 0 typical

Therefore, in this approximation, the pressure dependence on the radius is piecewise: parabolic in the core (Equation (21)) and logarithmic in the mantle (Equation (14)), and they interconnect at CMB. This piecewise pressure proﬁle can be closely matched to the realistic pressure proﬁle calculated from PREM as shown in Figure 2 Panel (3)a-d. Equation (22)

tends to signiﬁcantly underestimate the P0 towards higher mass and higher CMF due to signiﬁcant core compression. The following semi-empirical formula (Equation (23)) corrects

this eﬀect. Tested against numerical simulations, it gives a better estimate of P0 to within ∼ 10% error in the range of CMF ∈ 0.1 ∼ 0.4 and mass ∈ 0.1 ∼ 10M⊕.

Mp Pf0 ≈ (200 + 600 · CMF) · GPa (23) M⊕

A better approximation for the core pressure proﬁle using Pf0 from Equation (23) and PCMB from Equation (15), shown as the purple curves in Figure 2 Panel (3)a-d, is

r 2 Pf0(r) = Pf0 − (Pf0 − PCMB) · (24) Rcore

The corrected core gravity g^core(r), shown as purple curves in Figure 2 Panel (1)a-d, and the corrected core density ρgcore, shown as the purple curves in Figure 2 Panel (2)a-d, can be calculated as from Pf0 as

v u ! g^core(r) ρcore u Pf0 − PCMB) = g = t (25) gcore(r) ρcore P0 − PCMB)

ρgcore from Equation (25) tends to better approximates the core density near the center, while ρcore from Equation (10a) approximates the core density near the CMB.

5. Energy of Core Formation

The energy of core formation can be estimated as the diﬀerence in gravitational energies between the uniform-density state and this simple analytical model. According to the Virial – 11 –

Theorem (Haswell 2010), the total gravitational energy is

Z surface P Egrav = −3 · dm (26) center ρ

With the analytic forms of P and ρ in this model, Equation (26) can be integrated to obtain

2 GMp 1 3 Egrav = − · 2 − · CRF (27) 3Rp 5

Comparing it to the gravitational energy of a uniform-density sphere, Egrav, uniform sphere = 2 − 3 GMp , the diﬀerence of the two can be regarded as the energy released during core formation 5 Rp (gravitational energy released from the concentration of denser materials toward the center):

2 2 GMp 1 3 GMp 1 3/2 Ediﬀ = Egrav, uniform sphere − Egrav = (1 − CRF ) = (1 − CMF ) Rp 15 Rp 15 (28) Since CMF ∈ 0.1 ∼ 0.4, the term CMF3/2 is small enough to be dropped to give

2 1 GMp 1 Ediﬀ ≈ ≈ | Egrav | (29) 15 Rp 10

Therefore, the energy released during core formation is ∼ 10% of the total gravitational 31 energy of such a rocky planet. For Earth, Ediﬀ,⊕ ≈ 2.5 ∗ 10 J. The calculated values of other planets are listed in Table 1.

6. Thermal Content of the Planet

Since the temperatures inside the mantle of such a planet are likely above the Debye temperature of the solid, the heat capacity per mole of the atoms can be approximated as 3R (gas constant R = 8.314 J K−1mol−1). The speciﬁc heat capacity (heat capacity per unit mass) is 3R/µ where µ is the average atomic weight of the composition, which for Mg-

silicates (MgO, SiO2, or any proportion of them combined, such as MgSiO3 or Mg2SiO4) is −1 0.02 kg mol . The speciﬁc thermal energy uth of the mantle material is thus – 12 –

3RT u = (30) th µ where T is temperature. The total thermal energy of mantle is calculated by the integration:

Z Mp Z 1 Eth,mantle = uth · dm = Mp · uth · dx (31) Mcore CMF where x ≡ m . In this model, the mantle density ρ ∝ 1 ∝ √1 (Equation (10b)). On the Mp mantle r m other hand, with the assumption of the adiabatic temperature gradient in the mantle and ∂ln(T ) the introduction of the Gr¨uneisenparameter γ ≡ ∂ln(ρ) |adiabat, the speciﬁc thermal energy can be rewritten to show its functional dependence on density ρ or mass m (Zeng & Jacobsen 2016):

γ γ − 2 3R · Tmp ρ 3R · Tmp m uth = · = · (32) µ ρ0 µ Mp where Tmp (mantle potential temperature) is deﬁned as the temperature where the mantle adiabat is extrapolated to zero pressure. For silicates, γ ∼ 1, then,

3R · T √ E = 2 · M · mp (1 − CMF) (33) th,mantle p µ √ For CMF ≈ 0.3, CMF ≈ CRF ≈ 0.5, and the total thermal energy of the mantle (considering only vibrations of the atoms in crystal lattices, while neglecting the electron contribution) is

3R · Tmp Tmp Mp 30 Eth,mantle ≈ · Mp ≈ · · 7 · 10 J (34) µ 1000K M⊕

Equation (34) suggests that concerning the thermal content, the mantle can be treated as an uncompressed mass of Mp at isothermal temperature of Tmp. Therefore, an eﬀective heat capacity Cth,mantle of the mantle can be deﬁned with respect to Tmp:

3R Mp 27 −1 Cth,mantle ≈ · Mp ≈ · 7 · 10 JK (35) µ M⊕

The detailed calculation in Stacey & Davis (2008) shows that the eﬀective heat capacity of the Earth’s mantle is 7.4 · 1027 JK−1, indeed close to our estimate. Since the core is smaller – 13 –

in comparison (CMF ∈ 0.1 ∼ 0.4) and the core material has smaller speciﬁc heat capacity than silicates, the thermal content of the core should be generally less than that of the mantle. The mantle shall dictate the cooling of the core (Stacey & Davis 2008). Therefore, the mantle heat capacity can be regarded as an approximation for the total heat capacity of a planet.

Due to a feedback mechanism of silicate melting, there is good reason to set Tmp ≈ 1600 K for Earth and super-Earths for a ﬁrst approximation (Stixrude 2014). Then the temperature proﬁles inside these planets can be estimated using formulae in Stixrude (2014). The results are listed in Table 1 and plotted in Figure 2 Panel (4)a-d. In general, the mantle adiabat has shallower slope than the mantle melting curves (Zeng & Jacobsen 2016), so the mantle is mostly solid, while the core is partially or fully molten.

7. Moment of Inertia

The moment of inertia is calculated from Equation (36), where x represents the distance of the mass element to the rotational axis and the integration is over the entire volume (V ):

ZZZ I = x2 · ρ(~r) · dV (36)

V Considering two simple cases:

( 2 2 thin spherical shell with radius Rp, Ishell = 3 MRp 2 2 uniform solid sphere with radius Rp, Isolid sphere = 5 MRp 2 2 Deﬁning C ≡ I/MRp as the moment-of-inertia factor, for shell C = 3 , and for sphere 2 C = 5 . Smaller C corresponds to the mass being more concentrated toward the center. For this model, the moment of inertia of the core and the mantle can each be calculated separately, then combined to give the total moment of inertia of the planet:

2 I = M R2 · CMF2 (37) core 5 p p

1 I = M R2 · (1 − CMF2) (38) mantle 3 p p

1 1 I = I + I = M R2 · 1 + · CMF2 (39) total core mantle 3 p p 5 – 14 –

1 2 1 Considering CMF∈ 0.1 ∼ 0.4, the term 5 CMF can be ignored, so C ≈ 3 . In the solar 1 system, the C for Mercury, Venus, and Earth is indeed very close to 3 (Rubie et al. 2007). 1 Here we show that C ≈ 3 can be generalized to other rocky exoplanets:

1 I ≈ · M · R2 (40) planet 3 p p

2 38 2 1 2 37 2 For Earth, M⊕R⊕ = 2.4 · 10 kg m , and I⊕ ≈ 3 M⊕R⊕ = 8 · 10 kg m . The angular 33 2 −1 momentum of Earth’s rotation is L⊕ = I⊕ · Ω⊕ = 6 · 10 kg m s . The total rotational 2 1 2 L⊕ 29 −5 −1 energy of Earth is Erot = · I⊕ · Ω⊕ = 2 = 2 · 10 J, where Ω⊕ = 7.3 · 10 rad s is the 2 2·Ω⊕ angular frequency of Earth’s rotation. The calculated moment of inertia of other planets are shown in Table 1.

8. Conclusion

A simple analytical model for rocky planetary interiors is presented here and compared to numerical results. It explores the scaling relations among the following ﬁve aspects: (1) the relative size and mass of the core and the mantle, (2) the interior pressure and gravity, (3) the core formation energy and the gravitational energy, (4) the heat content and temperature proﬁles, and (5) the moment of inertia. Other results can be derived from this model. Although being approximate, these results are straightforward to apply, as in many cases mass and radius are only measured approximately. Combined with the mass-radius relation, these formulae shall provide us with a new way of looking at the rocky planetary interiors, complementing the numerical approach.

9. Acknowledgement

This work was supported by a grant from the Simons Foundation (SCOL [award #337090] to L.Z.). The authors would like to thank Mr. and Mrs. Simons and the Simons Foundation for generously supporting this research project. The author L.Z. would like to thank his father Lingwei Zeng and mother Ling Li for continuous support and help. He also would like to thank Eugenia Hyung for insightful discussion of this paper. Part of this research was also conducted under the Sandia Z Fundamental Science Program and supported by the Department of Energy National Nuclear Security Administration under Award Numbers DE-NA0001804 and DE-NA0002937 to S.B.J (PI) with Harvard University. This research – 15 – reﬂects the authors’ views and not those of the DOE. Finally, the authors would like to thank Dimitar D. Sasselov for insightful suggestions and helpful comments on this paper. – 16 –

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This preprint was prepared with the AAS LATEX macros v5.2.